Classifying wave forecasts with model-based Geostatistics and the Aitchison distribution

Abstract

This paper proposes a non-parametric method of classification of maps (i.e., variable fields such as wave energy maps for the Western Mediterranean Sea) into a set of D typical regimes (calm, E-, SW- or N/NW-wind dominated storms, the 4 synoptic situations more often occurring in this region). Each map in the training set is described by its values at P measurement points and one of these regime classes. A map is thus identified as a labelled point in a P-dimensional feature space, and the problem is to find a discrimination rule that may be used for attaching a classification probability to future unlabelled maps. The discriminant model proposed assumes that some log-contrasts of these classification probabilities form a Gaussian random field on the feature space. Then, available data (labelled maps of the training set) are linked to these latent probabilities through a multinomial model. This model is quite common in model-based Geostatistics and the Gaussian process classification literature. Inference is here approximated numerically using likelihood based techniques. The multinomial likelihood of labelled features is combined in a Bayesian updating with the Gaussian random field, playing the role of prior distribution. The posterior corresponds to an Aitchison distribution. Its maximum posterior estimates are obtained in two steps, exploiting several properties of this family. The first step is to obtain the mode of this distribution for labelled features, by solving a mildly non-linear system of equations. The second step is to propagate these estimates to unlabelled features, with simple kriging of log-contrasts. These inference steps can be extended via Markov-chain Monte Carlo (MCMC) sampling to a hierarchical Bayesian problem. This MCMC sampling can be improved by further exploiting the Aitchison distribution properties, though this is only outlined here. Results for the application case study suggest that E- and N/NW-dominated storms can be successfully discriminated from calm situations, but not so easily distinguished from each other.